Chapter 1
Sampling and Descriptive Statistics
Introduction
The collection and analysis of data are fundamental to science and engineering. Scientists
discover the principles that govern the physical world, and engineers learn how to design
important new products and processes, by analyzing data collected in scientific experiments. A
major difficulty with scientific data is that they are subject to random variation, or uncertainty.
That is, when scientific measurements are repeated, they come out somewhat differently each
time. This poses a problem: How can one draw conclusions from the results of an experiment
when those results could have come out differently? To address this question, a knowledge of
statistics is essential. Statistics is the field of study concerned with the collection, analysis, and
interpretation of uncertain data. The methods of statistics allow scientists and engineers to design
valid experiments and to draw reliable conclusions from the data they produce.
Although our emphasis in this book is on the applications of statistics to science and
engineering, it is worth mentioning that the analysis and interpretation of data are playing an
ever-increasing role in all aspects of modern life. For better or worse, huge amounts of data are
collected about our opinions and our lifestyles, for purposes ranging from the creation of more
effective marketing campaigns to the development of social policies designed to improve our
way of life. On almost any given day, newspaper articles are published that purport to explain
social or economic trends through the analysis of data. A basic knowledge of statistics is
therefore necessary not only to be an effective scientist or engineer, but also to be a wellinformed member of society.
The Basic Idea
The basic idea behind all statistical methods of data analysis is to make inferences about a
population by studying a relatively small sample chosen from it. As an illustration, Page 2
consider a machine that makes steel rods for use in optical storage devices. The
specification for the diameter of the rods is 0.45 ± 0.02 cm. During the last hour, the machine has
made 1000 rods. The quality engineer wants to know approximately how many of these rods
meet the specification. He does not have time to measure all 1000 rods. So he draws a random
sample of 50 rods, measures them, and finds that 46 of them (92%) meet the diameter
specification. Now, it is unlikely that the sample of 50 rods represents the population of 1000
perfectly. The proportion of good rods in the population is likely to differ somewhat from the
sample proportion of 92%. What the engineer needs to know is just how large that difference is
likely to be. For example, is it plausible that the population percentage could be as high as 95%?
98%? As low as 90%? 85%?
Here are some specific questions that the engineer might need to answer on the basis of
these sample data:
1. The engineer needs to compute a rough estimate of the likely size of the difference between
the sample proportion and the population proportion. How large is a typical difference for
this kind of sample?
2. The quality engineer needs to note in a logbook the percentage of acceptable rods
manufactured in the last hour. Having observed that 92% of the sample rods were good, he
will indicate the percentage of acceptable rods in the population as an interval of the form
92% ± x%, where x is a number calculated to provide reasonable certainty that the true
population percentage is in the interval. How should x be calculated?
3. The engineer wants to be fairly certain that the percentage of good rods is at least 90%;
otherwise he will shut down the process for recalibration. How certain can he be that at least
90% of the 1000 rods are good?
Much of this book is devoted to addressing questions like these. The first of these questions
requires the computation of a standard deviation, which we will discuss in Chapters 2 and 4.
The second question requires the construction of a confidence interval, which we will learn
about in Chapter 5. The third calls for a hypothesis test, which we will study in Chapter 6.
The remaining chapters in the book cover other important topics. For example, the engineer
in our example may want to know how the amount of carbon in the steel rods is related to their
tensile strength. Issues like this can be addressed with the methods of correlation and
regression, which are covered in Chapters 7 and 8. It may also be important to determine how to
adjust the manufacturing process with regard to several factors, in order to produce optimal
results. This requires the design of factorial experiments, which are discussed in Chapter 9.
Finally, the engineer will need to develop a plan for monitoring the quality of the product
manufactured by the process. Chapter 10 covers the topic of statistical quality control, in which
statistical methods are used to maintain quality in an industrial setting.
The topics listed here concern methods of drawing conclusions from data. These methods
form the field of inferential statistics. Before we discuss these topics, we must first learn more
about methods of collecting data and of summarizing clearly the basic information they contain.
These are the topics of sampling and descriptive statistics, and they are covered in the rest of
this chapter.
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1-1
Sampling
As mentioned, statistical methods are based on the idea of analyzing a sample drawn from a
population. For this idea to work, the sample must be chosen in an appropriate way. For
example, let us say that we wished to study the heights of students at the Colorado School of
Mines by measuring a sample of 100 students. How should we choose the 100 students to
measure? Some methods are obviously bad. For example, choosing the students from the rosters
of the football and basketball teams would undoubtedly result in a sample that would fail to
represent the height distribution of the population of students. You might think that it would be
reasonable to use some conveniently obtained sample, for example, all students living in a
certain dorm or all students enrolled in engineering statistics. After all, there is no reason to think
that the heights of these students would tend to differ from the heights of students in general.
Samples like this are not ideal, however, because they can turn out to be misleading in ways that
are not anticipated. The best sampling methods involve random sampling. There are many
different random sampling methods, the most basic of which is simple random sampling.
To understand the nature of a simple random sample, think of a lottery. Imagine that 10,000
lottery tickets have been sold and that 5 winners are to be chosen. What is the fairest way to
choose the winners? The fairest way is to put the 10,000 tickets in a drum, mix them thoroughly,
and then reach in and one by one draw 5 tickets out. These 5 winning tickets are a simple random
sample from the population of 10,000 lottery tickets. Each ticket is equally likely to be one of the
5 tickets drawn. More importantly, each collection of 5 tickets that can be formed from the
10,000 is equally likely to be the group of 5 that is drawn. It is this idea that forms the basis for
the definition of a simple random sample.
Summary
■
■
■
A population is the entire collection of objects or outcomes about which information is
sought.
A sample is a subset of a population, containing the objects or outcomes that are actually
observed.
A simple random sample of size n is a sample chosen by a method in which each
collection of n population items is equally likely to make up the sample, just as in a
lottery.
Since a simple random sample is analogous to a lottery, it can often be drawn by the same
method now used in many lotteries: with a computer random number generator. Suppose there
are N items in the population. One assigns to each item in the population an integer between 1
and N. Then one generates a list of random integers between 1 and N and chooses the
corresponding population items to make up the simple random sample.
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Example
1.1
A physical education professor wants to study the physical fitness levels of students at her
university. There are 20,000 students enrolled at the university, and she wants to draw a
sample of size 100 to take a physical fitness test. She obtains a list of all 20,000 students,
numbered from 1 to 20,000. She uses a computer random number generator to generate 100
random integers between 1 and 20,000 and then invites the 100 students corresponding to
those numbers to participate in the study. Is this a simple random sample?
Solution
Yes, this is a simple random sample. Note that it is analogous to a lottery in which each
student has a ticket and 100 tickets are drawn.
Example
1.2
A quality engineer wants to inspect rolls of wallpaper in order to obtain information on the
rate at which flaws in the printing are occurring. She decides to draw a sample of 50 rolls of
wallpaper from a day's production. Each hour for 5 hours, she takes the 10 most recently
produced rolls and counts the number of flaws on each. Is this a simple random sample?
Solution
No. Not every subset of 50 rolls of wallpaper is equally likely to make up the sample. To
construct a simple random sample, the engineer would need to assign a number to each roll
produced during the day and then generate random numbers to determine which rolls make up
the sample.
In some cases, it is difficult or impossible to draw a sample in a truly random way. In these
cases, the best one can do is to sample items by some convenient method. For example, imagine
that a construction engineer has just received a shipment of 1000 concrete blocks, each weighing
approximately 50 pounds. The blocks have been delivered in a large pile. The engineer wishes to
investigate the crushing strength of the blocks by measuring the strengths in a sample of 10
blocks. To draw a simple random sample would require removing blocks from the center and
bottom of the pile, which might be quite difficult. For this reason, the engineer might construct a
sample simply by taking 10 blocks off the top of the pile. A sample like this is called a sample of
convenience.
Definition
A sample of convenience is a sample that is obtained in some convenient way, and not drawn
by a well-defined random method.
The big problem with samples of convenience is that they may differ systematically in some
way from the population. For this reason samples of convenience should not be used, except in
situations where it is not feasible to draw a random sample. When it is necessary to take a Page 5
sample of convenience, it is important to think carefully about all the ways in which the
sample might differ systematically from the population. If it is reasonable to believe that no
important systematic difference exists, then it may be acceptable to treat the sample of
convenience as if it were a simple random sample. With regard to the concrete blocks, if the
engineer is confident that the blocks on the top of the pile do not differ systematically in any
important way from the rest, then he may treat the sample of convenience as a simple random
sample. If, however, it is possible that blocks in different parts of the pile may have been made
from different batches of mix or may have different curing times or temperatures, a sample of
convenience could give misleading results.
Some people think that a simple random sample is guaranteed to reflect its population
perfectly. This is not true. Simple random samples always differ from their populations in some
ways, and occasionally may be substantially different. Two different samples from the same
population will differ from each other as well. This phenomenon is known as sampling
variation. Sampling variation is one of the reasons that scientific experiments produce
somewhat different results when repeated, even when the conditions appear to be identical.
Example
1.3
A quality inspector draws a simple random sample of 40 bolts from a large shipment and
measures the length of each. He finds that 34 of them, or 85%, meet a length specification. He
concludes that exactly 85% of the bolts in the shipment meet the specification. The inspector's
supervisor concludes that the proportion of good bolts is likely to be close to, but not exactly
equal to, 85%. Which conclusion is appropriate?
Solution
Because of sampling variation, simple random samples don't reflect the population perfectly.
They are often fairly close, however. It is therefore appropriate to infer that the proportion of
good bolts in the lot is likely to be close to the sample proportion, which is 85%. It is not
likely that the population proportion is equal to 85%, however.
Example
1.4
Continuing Example 1.3, another inspector repeats the study with a different simple random
sample of 40 bolts. She finds that 36 of them, or 90%, are good. The first inspector claims that
she must have done something wrong, since his results showed that 85%, not 90%, of bolts
are good. Is he right?
Solution
No, he is not right. This is sampling variation at work. Two different samples from the same
population will differ from each other and from the population.
Since simple random samples don't reflect their populations perfectly, why is it important
that sampling be done at random? The benefit of a simple random sample is that there is no
systematic mechanism tending to make the sample unrepresentative. The differences Page 6
between the sample and its population are due entirely to random variation. Since the
mathematical theory of random variation is well understood, we can use mathematical models to
study the relationship between simple random samples and their populations. For a sample not
chosen at random, there is generally no theory available to describe the mechanisms that caused
the sample to differ from its population. Therefore, nonrandom samples are often difficult to
analyze reliably.
In Examples 1.1 to 1.4, the populations consisted of actual physical objects—the students at
a university, the concrete blocks in a pile, the bolts in a shipment. Such populations are called
tangible populations. Tangible populations are always finite. After an item is sampled, the
population size decreases by 1. In principle, one could in some cases return the sampled item to
the population, with a chance to sample it again, but this is rarely done in practice.
Engineering data are often produced by measurements made in the course of a scientific
experiment, rather than by sampling from a tangible population. To take a simple example,
imagine that an engineer measures the length of a rod five times, being as careful as possible to
take the measurements under identical conditions. No matter how carefully the measurements are
made, they will differ somewhat from one another, because of variation in the measurement
process that cannot be controlled or predicted. It turns out that it is often appropriate to consider
data like these to be a simple random sample from a population. The population, in these cases,
consists of all the values that might possibly have been observed. Such a population is called a
conceptual population, since it does not consist of actual objects.
A simple random sample may consist of values obtained from a process under identical
experimental conditions. In this case, the sample comes from a population that consists of
all the values that might possibly have been observed. Such a population is called a
conceptual population.
Example 1.5 involves a conceptual population.
Example
1.5
A geologist weighs a rock several times on a sensitive scale. Each time, the scale gives a
slightly different reading. Under what conditions can these readings be thought of as a simple
random sample? What is the population?
Solution
If the physical characteristics of the scale remain the same for each weighing, so that the
measurements are made under identical conditions, then the readings may be considered to be
a simple random sample. The population is conceptual. It consists of all the readings that the
scale could in principle produce.
Note that in Example 1.5, it is the physical characteristics of the measurement process that
determine whether the data are a simple random sample. In general, when deciding Page 7
whether a set of data may be considered to be a simple random sample, it is necessary to have
some understanding of the process that generated the data. Statistical methods can sometimes
help, especially when the sample is large, but knowledge of the mechanism that produced the
data is more important.
Example
1.6
A new chemical process has been designed that is supposed to produce a higher yield of a
certain chemical than does an old process. To study the yield of this process, we run it 50
times and record the 50 yields. Under what conditions might it be reasonable to treat this as a
simple random sample? Describe some conditions under which it might not be appropriate to
treat this as a simple random sample.
Solution
To answer this, we must first specify the population. The population is conceptual and
consists of the set of all yields that will result from this process as many times as it will ever
be run. What we have done is to sample the first 50 yields of the process. If, and only if, we
are confident that the first 50 yields are generated under identical conditions, and that they do
not differ in any systematic way from the yields of future runs, then we may treat them as a
simple random sample.
Be cautious, however. There are many conditions under which the 50 yields could fail to
be a simple random sample. For example, with chemical processes, it is sometimes the case
that runs with higher yields tend to be followed by runs with lower yields, and vice versa.
Sometimes yields tend to increase over time, as process engineers learn from experience how
to run the process more efficiently. In these cases, the yields are not being generated under
identical conditions and would not be a simple random sample.
Example 1.6 shows once again that a good knowledge of the nature of the process under
consideration is important in deciding whether data may be considered to be a simple random
sample. Statistical methods can sometimes be used to show that a given data set is not a simple
random sample. For example, sometimes experimental conditions gradually change over time. A
simple but effective method to detect this condition is to plot the observations in the order they
were taken. A simple random sample should show no obvious pattern or trend.
Figure 1.1 (page 8) presents plots of three samples in the order they were taken. The plot in
Figure 1.1a shows an oscillatory pattern. The plot in Figure 1.1b shows an increasing trend.
Neither of these samples should be treated as a simple random sample. The plot in Figure 1.1c
does not appear to show any obvious pattern or trend. It might be appropriate to treat these data
as a simple random sample. However, before making that decision, it is still important to think
about the process that produced the data, since there may be concerns that don't show up in the
plot (see Example 1.7).
FIGURE 1.1
Three plots of observed values versus the order in which they were made. (a) The values
show a definite pattern over time. This is not a simple random sample. (b) The values show a trend over
time. This is not a simple random sample. (c) The values do not show a pattern or trend. It may be
appropriate to treat these data as a simple random sample.
Sometimes the question as to whether a data set is a simple random sample depends on the
population under consideration. This is one case in which a plot can look good, yet the data are
not a simple random sample. Example 1.7 provides an illustration.
Page 8
Example
1.7
A new chemical process is run 10 times each morning for five consecutive mornings. A plot
of yields in the order they are run does not exhibit any obvious pattern or trend. If the new
process is put into production, it will be run 10 hours each day, from 7 A.M. until 5 P.M. Is it
reasonable to consider the 50 yields to be a simple random sample? What if the process will
always be run in the morning?
Solution
Since the intention is to run the new process in both the morning and the afternoon, the
population consists of all the yields that would ever be observed, including both morning and
afternoon runs. The sample is drawn only from that portion of the population that consists of
morning runs, and thus it is not a simple random sample. There are many things that could go
wrong if this is used as a simple random sample. For example, ambient temperatures may
differ between morning and afternoon, which could affect yields.
If the process will be run only in the morning, then the population consists only of
morning runs. Since the sample does not exhibit any obvious pattern or trend, it might well be
appropriate to consider it to be a simple random sample.
Independence
The items in a sample are said to be independent if knowing the values of some of them does
not help to predict the values of the others. With a finite, tangible population, the items in a
simple random sample are not strictly independent, because as each item is drawn, the population
changes. This change can be substantial when the population is small. However, when the
population is very large, this change is negligible and the items can be treated as if they were
independent.
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To illustrate this idea, imagine that we draw a simple random sample of 2 items
from the population
For the first draw, the numbers 0 and 1 are equally likely. But the value of the second item is
clearly influenced by the first; if the first is 0, the second is more likely to be 1, and vice versa.
Thus the sampled items are dependent. Now assume we draw a sample of size 2 from this
population:
Again on the first draw, the numbers 0 and 1 are equally likely. But unlike the previous example,
these two values remain almost equally likely on the second draw as well, no matter what
happens on the first draw. With the large population, the sample items are for all practical
purposes independent.
It is reasonable to wonder how large a population must be in order that the items in a simple
random sample may be treated as independent. A rule of thumb is that when sampling from a
finite population, the items may be treated as independent so long as the sample contains 5% or
less of the population.
Interestingly, it is possible to make a population behave as though it were infinitely large,
by replacing each item after it is sampled. This method is called sampling with replacement.
With this method, the population is exactly the same on every draw and the sampled items are
truly independent.
With a conceptual population, we require that the sample items be produced under identical
experimental conditions. In particular, then, no sample value may influence the conditions under
which the others are produced. Therefore, the items in a simple random sample from a
conceptual population may be treated as independent. We may think of a conceptual population
as being infinite, or equivalently, that the items are sampled with replacement.
Summary
■
■
The items in a sample are independent if knowing the values of some of the items does
not help to predict the values of the others.
Items in a simple random sample may be treated as independent in many cases
encountered in practice. The exception occurs when the population is finite and the
sample consists of a substantial fraction (more than 5%) of the population.
Other Sampling Methods
Page 10
In addition to simple random sampling there are other sampling methods that are useful in
various situations. In weighted sampling, some items are given a greater chance of being
selected than others, like a lottery in which some people have more tickets than others. In
stratified random sampling, the population is divided up into subpopulations, called strata,
and a simple random sample is drawn from each stratum. In cluster sampling, items are drawn
from the population in groups, or clusters. Cluster sampling is useful when the population is too
large and spread out for simple random sampling to be feasible. For example, many U.S.
government agencies use cluster sampling to sample the U.S. population to measure sociological
factors such as income and unemployment. A good source of information on sampling methods
is Scheaffer, et al., (2012).
Simple random sampling is not the only valid method of random sampling. But it is the
most fundamental, and we will focus most of our attention on this method. From now on, unless
otherwise stated, the terms “sample” and “random sample” will be taken to mean “simple
random sample.”
Types of Experiments
There are many types of experiments that can be used to generate data. We briefly describe a few
of them. In a one-sample experiment, there is only one population of interest, and a single
sample is drawn from it. For example, imagine that a process is being designed to produce
polyethylene that will be used to line pipes. An experiment in which several specimens of
polyethylene are produced by this process, and the tensile strength of each is measured, is a onesample experiment. The measured strengths are considered to be a simple random sample from a
conceptual population of all the possible strengths that can be observed for specimens
manufactured by this process. One-sample experiments can be used to determine whether a
process meets a certain standard, for example, whether it provides sufficient strength for a given
application.
In a multisample experiment, there are two or more populations of interest, and a sample is
drawn from each population. For example, if several competing processes are being considered
for the manufacture of polyethylene, and tensile strengths are measured on a sample of
specimens from each process, this is a multisample experiment. Each process corresponds to a
separate population, and the measurements made on the specimens from a particular process are
considered to be a simple random sample from that population. The usual purpose of
multisample experiments is to make comparisons among populations. In this example, the
purpose might be to determine which process produced the greatest strength or to determine
whether there is any difference in the strengths of polyethylene made by the different processes.
In many multisample experiments, the populations are distinguished from one another by
the varying of one or more factors that may affect the outcome. Such experiments are called
factorial experiments. For example, in his M.S. thesis at the Colorado School of Mines, G.
Fredrickson measured the Charpy V-notch impact toughness for a large number of welds. Each
weld was made with one of two types of base metals and had its toughness measured at one of
several temperatures. This was a factorial experiment with two factors: base metal and
temperature. The data consisted of several toughness measurements made at each Page 11
combination of base metal and temperature. In a factorial experiment, each combination
of the factors for which data are collected defines a population, and a simple random sample is
drawn from each population. The purpose of a factorial experiment is to determine how varying
the levels of the factors affects the outcome being measured. In his experiment Fredrickson
found that for each type of base metal, the toughness remained unaffected by temperature unless
the temperature was very low—below −100°C. As the temperature was decreased from −100°C
to −200°C, the toughness dropped steadily.
Types of Data
When a numerical quantity designating how much or how many is assigned to each item in a
sample, the resulting set of values is called numerical or quantitative. In some cases, sample
items are placed into categories, and category names are assigned to the sample items. Then the
data are categorical or qualitative. Example 1.8 provides an illustration.
Example
1.8
The article “Wind-Uplift Capacity of Residential Wood Roof-Sheathing Panels Retrofitted
with Insulating Foam Adhesive” (P. Datin, D. Prevatt, and W. Pang, Journal of Architectural
Engineering, 2011:144–154) presents tests in which air pressure was applied to roof-sheathing
panels until failure. The pressure at failure for each panel was recorded, along with the type of
sheathing, sheathing thickness, and wood species. The following table presents results of four
tests.
Sheathing
Failure
Type
Pressure (kPa)
5-ply plywood
2.63
Oriental Strand
3.69
Board
Cox plywood
5.26
4-ply plywood
5.03
Thickness
(mm)
Wood Species
11.9
Douglas Fir
15.1
Spruce-PineFir
12.7
Southern
Yellow Pine
15.9
Douglas Fir
Which data are numerical and which are categorical?
Solution
The failure pressure and thickness are numerical. The sheathing type and wood species are
categorical.
Controlled Experiments and Observational Studies
Many scientific experiments are designed to determine the effect of changing one or more factors
on the value of a response. For example, suppose that a chemical engineer wants to determine
how the concentrations of reagent and catalyst affect the yield of a process. The engineer can run
the process several times, changing the concentrations each time, and compare the yields that
result. This sort of experiment is called a controlled experiment, because the values of Page 12
the factors, in this case the concentrations of reagent and catalyst, are under the control
of the experimenter. When designed and conducted properly, controlled experiments can
produce reliable information about cause-and-effect relationships between factors and response.
In the yield example just mentioned, a well-done experiment would allow the experimenter to
conclude that the differences in yield were caused by differences in the concentrations of reagent
and catalyst.
There are many situations in which scientists cannot control the levels of the factors. For
example, there have been many studies conducted to determine the effect of cigarette smoking on
the risk of lung cancer. In these studies, rates of cancer among smokers are compared with rates
among non-smokers. The experimenters cannot control who smokes and who doesn't; people
cannot be required to smoke just to make a statistician's job easier. This kind of study is called an
observational study, because the experimenter simply observes the levels of the factor as they
are, without having any control over them. Observational studies are not nearly as good as
controlled experiments for obtaining reliable conclusions regarding cause and effect. In the case
of smoking and lung cancer, for example, people who choose to smoke may not be
representative of the population as a whole, and may be more likely to get cancer for other
reasons. For this reason, although it has been known for a long time that smokers have higher
rates of lung cancer than non-smokers, it took many years of carefully done observational studies
before scientists could be sure that smoking was actually the cause of the higher rate.
Exercises for Section 1.1
1.
2.
3.
Each of the following processes involves sampling from a population. Define the
population, and state whether it is tangible or conceptual.
a. A chemical process is run 15 times, and the yield is measured each time.
b. A pollster samples 1000 registered voters in a certain state and asks them which
candidate they support for governor.
c. In a clinical trial to test a new drug that is designed to lower cholesterol, 100 people
with high cholesterol levels are recruited to try the new drug.
d. Eight concrete specimens are constructed from a new formulation, and the compressive
strength of each is measured.
e. A quality engineer needs to estimate the percentage of bolts manufactured on a certain
day that meet a strength specification. At 3:00 in the afternoon he samples the last 100
bolts to be manufactured.
If you wanted to estimate the mean height of all the students at a university, which one of
the following sampling strategies would be best? Why? Note that none of the methods are
true simple random samples.
i. Measure the heights of 50 students found in the gym during basketball intramurals.
ii. Measure the heights of all engineering majors.
iii. Measure the heights of the students selected by choosing the first name on each page
of the campus phone book.
True or false:
A simple random sample is guaranteed to reflect exactly the population from which it
was drawn.
b. A simple random sample is free from any systematic tendency to differ from the
population from which it was drawn.
A sample of 100 college students is selected from all students registered at a certain college,
and it turns out that 38 of them participate in intramural sports. True or false:
a. The proportion of students at this college who participate in intramural sports is 0.38.
b. The proportion of students at this college who participate in intramural sports is likely
to be close to 0.38, but not equal to 0.38.
a.
4.
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5.
6.
7.
8.
9.
A certain process for manufacturing integrated circuits has been in use for a period of time,
and it is known that 12% of the circuits it produces are defective. A new process that is
supposed to reduce the proportion of defectives is being tested. In a simple random sample
of 100 circuits produced by the new process, 12 were defective.
a. One of the engineers suggests that the test proves that the new process is no better than
the old process, since the proportion of defectives in the sample is the same. Is this
conclusion justified? Explain.
b. Assume that there had been only 11 defective circuits in the sample of 100. Would this
have proven that the new process is better? Explain.
c. Which outcome represents stronger evidence that the new process is better: finding 11
defective circuits in the sample, or finding 2 defective circuits in the sample?
Refer to Exercise 5. True or false:
a. If the proportion of defectives in the sample is less than 12%, it is reasonable to
conclude that the new process is better.
b. If the proportion of defectives in the sample is only slightly less than 12%, the
difference could well be due entirely to sampling variation, and it is not reasonable to
conclude that the new process is better.
c. If the proportion of defectives in the sample is a lot less than 12%, it is very unlikely
that the difference is due entirely to sampling variation, so it is reasonable to conclude
that the new process is better.
To determine whether a sample should be treated as a simple random sample, which is more
important: a good knowledge of statistics, or a good knowledge of the process that produced
the data?
A medical researcher wants to determine whether exercising can lower blood pressure. At a
health fair, he measures the blood pressure of 100 individuals, and interviews them about
their exercise habits. He divides the individuals into two categories: those whose typical
level of exercise is low, and those whose level of exercise is high.
a. Is this a controlled experiment or an observational study?
b. The subjects in the low exercise group had considerably higher blood pressure, on the
average, than subjects in the high exercise group. The researcher concludes that
exercise decreases blood pressure. Is this conclusion well-justified? Explain.
A medical researcher wants to determine whether exercising can lower blood pressure. She
recruits 100 people with high blood pressure to participate in the study. She assigns a
random sample of 50 of them to pursue an exercise program that includes daily swimming
and jogging. She assigns the other 50 to refrain from vigorous activity. She measures the
blood pressure of each of the 100 individuals both before and after the study.
a. Is this a controlled experiment or an observational study?
b. On the average, the subjects in the exercise group substantially reduced their blood
pressure, while the subjects in the no-exercise group did not experience a reduction.
The researcher concludes that exercise decreases blood pressure. Is this conclusion
better justified than the conclusion in Exercise 8? Explain.